Integer Partitions and Convexity

Let n be an integer >=1, and let p(n,k) and
P(n,k) count the number of partitions
of n into k parts, and the number of partitions of n
into parts less than or equal to k, respectively.
In this paper, we show that these
functions are convex. The result includes the actual value of the
constant of Bateman and Erdős.